open import 1Lab.Path.Reasoning
open import 1Lab.Prelude

open import Algebra.Group.Cat.Base
open import Algebra.Group.Concrete
open import Algebra.Group.Ab

open import Cat.Prelude

open import Homotopy.Space.Delooping

module FirstGroupCohomology where

open Precategory

π₁BG≡G :  {} (G : Group )  π₁B (Concrete G)  G
π₁BG≡G G = π₁B≡π₀₊₁ (Concrete G)  sym (G≡π₁B G)

-- Any two loops commute in the delooping of an abelian group.
ab→square :  {} {H : Group } (H-ab : is-commutative-group H)
           {x : Deloop H} (p q : x  x)  Square p q q p
ab→square {H = H} H-ab {x} = Deloop-elim-prop H  x  (p q : x  x)  Square p q q p) hlevel!
   p q  commutes→square (subst is-commutative-group (sym (π₁BG≡G H)) H-ab p q)) x

module _ {} (G : Group ) (H : Group ) (H-ab : is-commutative-group H) where
  -- The first cohomology of G with coefficients in H.
  -- We will show that it is equivalent to the set of group homomorphisms from G
  -- to H, assuming that H is abelian.
  H¹[G,H] =  (Deloop G  Deloop H) ∥₀

  unpoint : (Deloop∙ G →∙ Deloop∙ H)  H¹[G,H]
  unpoint (f , _) = inc f

  work :  f  f base  base  is-contr (fibre unpoint (inc f))
  work f ptf .centre = (f , ptf) , refl
  work f ptf .paths ((g , ptg) , g≡f) = Σ-prop-path! (Σ-pathp
    (funext (Deloop-elim-set G _ hlevel! (ptf  sym ptg) λ z  ∥-∥-rec!
       g≡f  J
         g _   ptg  Square (ap f (path z)) (ptf  sym ptg) (ptf  sym ptg) (ap g (path z)))
         _  ab→square H-ab _ _)
        (sym g≡f) ptg)
      (∥-∥₀ g≡f)))
    (flip₂ (∙-filler'' ptf (sym ptg))))

  unpoint-is-equiv : is-equiv unpoint
  unpoint-is-equiv .is-eqv = ∥-∥₀-elim  _  hlevel 2)
    λ f  ∥-∥-rec! (work f) (Deloop-is-connected (f base))

  unpoint≃ : H¹[G,H]  (Deloop∙ G →∙ Deloop∙ H)
  unpoint≃ = (unpoint , unpoint-is-equiv) e⁻¹

  delooping : (Deloop∙ G →∙ Deloop∙ H)  Hom (Groups ) (π₁B (Concrete G)) (π₁B (Concrete H))
  delooping = _ , π₁F-is-ff {_} {Concrete G} {Concrete H}

  first-group-cohomology : H¹[G,H]  Hom (Groups ) G H
  first-group-cohomology = unpoint≃ ∙e delooping
    ∙e path→equiv (ap₂ (Hom (Groups )) (π₁BG≡G G) (π₁BG≡G H))

-- As a cool application, the space of endomorphisms of the delooping of ℤ/2ℤ has
-- exactly two connected components!
-- (But note that there is no type with exactly two endomorphisms: it would be a set,
-- and nⁿ = 2 has no integer solutions.)